Difference Between Tensor and Tensor field?

The difference is all in your head. Literally.

The difference in calling the same object $A$ a "tensor over $\mathfrak{X}(M)$" as opposed to "a tensor field over $M$" is that the former emphasizes the fact that we have an algebraic object: a tensor over some module, while the latter emphasizes the fact that underlying the module there is some manifold and geometry is going on there.

Calling something a tensor field instead of a tensor forces you to remember that $\mathfrak{X}(M)$ is not just some arbitrary module, but that its elements can be identified with smooth sections of the tangent bundle of some manifold. These additional structures are occasionally useful.